Suppose f⃗ (x, y)=(3x−4y)i⃗ +5xj⃗ and c is the counter-clockwise oriented sector of a circle centered at the origin with radius 3 and central angle π/3. use green's theorem to calculate the circulation of f⃗ around c.

Prove (a) cosh2(x) − sinh2(x) = 1 and (b) 1 − tanh 2(x) = sech 2(x). solution (a) cosh2(x) − sinh2(x) = ex + e−x 2 2 − 2 = e2x + 2 + e−2x 4 − = 4 = . (b) we start with the identity proved in part (a): cosh2(x) − sinh2(x) = 1. if we divide both sides by cosh2(x), we get 1 − sinh2(x) cosh2(x) = 1 or 1 − tanh 2(x) = .

Jaxon has s packs of 8 crazy straws. he divided the crazy straws into 2 equal bags. now, he has 224 crazy straws in each bag. how many packs of crazy straws did jaxon start with?

Rachel is studying the population of a particular bird species in a national park. she observes every 10th bird of that species that she can find in the park. her conclusion is that the birds living in the park who belong to that species generally feed on insects. rachel's sample is . based on the sample she picked, rachel's generalization is